Beyond Biased Reptation

3.1. General Remarks

The biased reptation model rather well describes the mobility of long DNA in agarose and acrylamide, and under a wide variety of pulsed-field conditions. A notable exception is FIGE, for which biased reptation leads to qualitatively wrong predictions. Experiments investigating electrophoretic motions on a molecular scale and computer simulations were also developed in order to better understand the mechanisms of PFGE. These studies, presented in more detail in several sections of this volume, revealed new features of DNA dynamics in gels, and raised further criticism of the biased reptation model. In this section, we recall this criticism, the experiments it is based upon, and the theoretical developments it promoted. The section is organized from a "theorist's point of view," i.e., in terms of molecular mechanism rather than experimental technique or historical priority.

3.2. Field Inversion, Orientation Overshoot, and Internal Modes

3.2.1. Field Inversion

The germinal work of Carle, Frank, and Olson in FIGE reported a very strong band inversion effect, in which chromosomes VIII to II of Saccharomyces Cerevisae (600-800 kbp), for instance, migrated about ten times more slowly than chromosomes I (=250 kbp) and XII (=2000 kbp) (61). Such a dramatic nonmonotonous behavior as a function of chain size is not always observed (62,66) in FIGE. In all cases, however, a minimum is observed when the mobility is plotted vs pulse times. This minimum, which is associated with an "antiresonance" of the field period with a molecular characteristic time of the chains, deepens and shifts to larger times (roughly linearly), when the DNA size is increased. It is not predicted by the biased reptation model.

3.2.2. Chain Orientation

The original biased reptation model (with constant tube length) proposes various predictions for the orientation of chains (14). In permanent regime, the orientation is expected to scale as I? for e < 1 and for e > 1, and to be independent of ATor sufficiently large chains. It also predicts that chains orient monotonously at the onset of the electric field, in a time scaling as NE~2 for e < 1 and NEfor e > 1 (14). Conversely, they are expected to loose their orientation at the arrest of the field, in a reptation time scaling as jV3.

Fluorescence polarization (99), linear dichroism (100-103), and electric birefringence (89,104-106) confirmed the orientation of chains by the electric field. The time scales for reaching the steady state (89,100-103) and this for complete disorientation (89) are in qualitative agreement with scaling predictions of the reptation theory. These experiments, however, also revealed qualitative discrepancies with the biased reptation predictions. In constant field, the orientation seems to continuously increase as TV1 beyond the size at which a saturation is predicted by the theory. There is also a strong transient overshoot of the orientation at the onset of the field, and a fast relaxation process just after field interruption, which cannot be accounted for by the sole orientation of the tube in the field (other more subtle "undershoots" were reported, but we let them aside here to focus on the most spectacular observations). These effects were attributed to internal modes of the chains, i.e., to the failure of the assumption of constant tube length. The comparison of the main relaxation times of the orientation with mobility measurements on a molecular scale (107,108) and with field inversion results (109) support the idea that the "antiresonance" of the mobilty observed in FIGE and the orientation transient overshoot have a common molecular origin.

3.2.3. Generalization of Biased Reptation: Internal Modes

Taking internal modes into account immediately introduces terrible complications in the theory of biased reptation: Equations must take into account the positions of all chain sections and their interactions. Even without field, this problem could only receive an approximate solution, known as Doi's length fluctuation theory (109). One of us proposed (32) a generalization of biased reptation based on this theory. Drastic approximations were necessary to achieve analytical solution, and the theory is only qualitative. The striking finding is that, when field-inversion is applied to a chain, the choices of the chain head can be considerably altered by thermally activated length fluctuations, if the time scale of these fluctuations is of the order of the pulse times. In this case, we have shown that the chains may have two locally stable conformations, the usual oriented state obtained in constant field, and a more collapsed state. The overall chain orientation, and the chain mobility as a consequence, are reduced by a factor that can reach 10. This negative resonance-type effect leads to a deep minimum in the mobility vs pulse time curve (Fig. 8), as experimentally observed. Our prediction for the evolution of this minimum with size, however, was smaller than the experimental one. In the light of further simulations and videomicroscopy experiements, we now believe that this discrepancy is attributable to the coupling of length fluctuations with the field, not considered in the model.

reverse ratio of times is 2, and the critical size N* is fixed to 10. Full lines correspond to the quantitative predictions in fully established dynamic regimes, the dotted lines correspond to the crossover regions where the model provides no quantitative prediction.

A rather opposite and complementary route was taken by Lim et al. (110), who modeled the DNA by a chain of non-Hookean springs (i.e., springs that cannot extend past a finite length corresponding to the fully extended length of the DNA). This model was solved analytically in the deterministic "preaveraged" approximation, i.e., without considering fluctuations. At the onset of the field, this chain of springs extends toward the field by its two ends, reaching a U-shaped very extended conformation. Then, it progressively slips around the bottom of the U like a rope on a pulley, and recover a more classical linear conformation. This analytical model describes well the experimentally observed orientation overshoot at the onset of the field, which is caused by this "pulley" effect. It does not, however, provide any insight into the mechanism of field-inversion, since it does not take into account fluctuations.

Lumpkin proposed a simpler model, consisting in only three beads with nonlinear drag connected by two springs (35). This is a minimal type of chain with internal modes, and obviously very far from actual long DNA chains, but it can be solved with less approximations than the previous ones. The analogy of this model with the electrophoresis of real DNA is rather indirect, but it is interesting that, in field-inversion conditions, such a simple model also leads to a sharp dip in the mobility vs pulse times.

Avery different theory was proposed by Zimm (5). The underlying model, which represents the tube as a sequence of wide "lakes" and narrow "straits" joining them, puts the emphasis on the fluctuations in tube diameter, whereas all previous reptation models assumed the tube diameter to be constant, or at most affected by a rather narrow distribution of pore sizes (92). Again, this model, which had to be solved numerically, presented a minimum in the mobility plotted as a function of pulse time. The evolution of the position of this minimum with chain length is in qualitative agreement with experiments. Its depth, however, seems too small.

3.2.4. Computer Simulations of the Generalized. Biased Reptation Model

Numerical simulations of the generalized biased reptation model (27,110) with internal modes have permitted progress beyond the limitations of analytical calculations. They did show that an extensible chain in a tube presents the following features:

1. Nonuniform extension, the head section being more compact than the tail.

2. An important overshoot of the extension and of the orientation at the onset of the field, owing to the U-shape effect already demonstrated by analytical calculations.

3. In agreement with experiment, distinct undershoots of the orientation when the field is reversed.

4. A minimum in the mobility as a function of pulse time in FIGE, which shifts to larger times when the size of the chain is increased.

The big disadvantage of these simulations as compared to analytical calculations, however, is that they have to be performed explicitly for a given chain size, and computation time did not allow the investigation of very long chains. In particular, it has not yet been possible to determine field-inversion mobilities for chains greater than the equivalent of about 50 kbp, far below the typical sizes at which strong band inversion is experimentally observed. For this reason, it is difficult to say if the weakness of the "mobility dip" observed in simulations is a consequence of the small size of the chains considered, or of a deeper inadequacy of the tube hypothesis.

Another way of simulating a chain in a tube with internal modes was recently proposed by Duke (36,37): it is a discretized version of generalized reptation, called "repton model" (111), in which the "blob" length can take values 0 or 1. This model is less time consuming than the previous one, so it can be applied to longer chains, but it remains limited to lower fields because of the discretization. Qualitatively, it leads to similar conclusions. Anticipating Section 3.3.1., it is interesting to notice that this repton model, which does not involve any tube leakage, presents the same type of "geometration" dynamics, which was associated by Deutsch with the presence of tube leakage in his own simulations.

3.3. Criticism of the Tube Assumption

In spite of its very attractive analytical simplicity, the tube hypothesis remains disputable in the presence of an external field (see discussion in Section 2.1.1.), and it was not universally accepted.

3.3.1. Simulations of a Chain Among Obstacles As early as 1986, Olivera de la Cruz (15), and then Deutsch and coworkers (21-24) and others (18-20) proposed a more direct approach, in which a charged flexible chain is simulated by Monte Carlo or Brownian Dynamics on a lattice of obstacles, without any tube assumption. These simulations permitted the investigation of mechanisms of motion more complicated than reptation for the first time. The most important is "tube leakage," i.e., formation of "hernias" (see Section 2.1.1.). These hernias seem to occur in several cases:

1. All along the chain, when the field is rotated by an angle around 90°, as in OFAGE (24).

2. At the chain's head in constant field. In that case, the competition between hernias apparently reduces the head's mobility, and induces a "bunching" of the chain. After a stretching time, which may be rather long, one of the hernias wins against the others, and stretch the chain in an aligned and reptative-like conformation again. Then, the motion of the chain is an alternation of strong collapsing and stretching events, called "geometration" by Deutsch (22), and rather different from the more continuous motion predicted by the biased reptation model. As quoted in the previous section, however, similar motions resulted from Duke's (36,37) simulation of a tube model without leakage, so they probably require some internal modes (flexibility) of the DNA chain, but not necessarily hernias.

3. In field inversion, finally, some pulsing frequencies induce a resonance between bunching events that occur at the two extremities of the chain, which strongly reduces the mobility and lead again to a "dip" in the mobility vs pulse time plot (22).

These simulations, more realistic than those of chains in a tube, are also more time-consuming, and they have not yet been able to provide accurate results on chains long enough to represent current pulsed electrophoresis situations (>100 kbp). Moreover, most of the simulations were performed in situations of high field (e of order 1). It would be interesting to know how the tube leakage and bunching processes evolve when the field is progressively lowered.

Finally, progress toward still more realism in the simulations at the expense of computing time have started, by means of molecular dynamics in three-dimensional lattices of obstacles (18,19) and on random percolation clusters of obstacles (20).

3.3.2. Discussion in the Light of Fluorescence Videomicroscopy

Considerable progress in the qualitative understanding of DNA migration in agarose, both in conventional and PFGE conditions, was recently promoted by fluorescence videomicroscopy experiments (75-77), by which individual molecules can be observed in real space and real time during their migration. These observations played in some sense the role of a "referee" in the field of theories and simulations. The first obvious observation is that, in average, the behavior of chains up to typically 200 kbp is in surprisingly good agreement with theoretical predictions and computer simulations:

1. The chains orient in the field, and orient more in higher fields, as predicted by the reptation model (1,2,4-14).

2. At the onset of the field, they stretch in U-shaped conformations, as expected from simulations (21-24) and analytical treatment of generalized reptation (110).

3. DNA seem to adopt two very different states of stretching, in agreement with theory (32) and simulations (21-24).

4. They also present a denser "head" and alternances of stretching and collapsing phases between these two stretching states, as predicted by simulations of lattice chains (21-24) and of the generalized reptation model (27,36,37).

5. The collapse of the head is often associated with the formation of hernias (i.e., loops), as in simulations in lattices of obstacles (21-24) and in contrast with tube models. These hernias lead to U-shaped conformations rather often in the course of migration.

6. A "resonance" in field inversion is also observed, in which the chain is "trapped" between two bunched extremities, in close resemblance with simulations (22,37).

7. As far as CFGE is concerned, two rather different behaviors have been reported: At rather low fields (3 V/cm) (76), DNA adopt for moderate angles "staircase" conformations equivalent to the "zigzag" model of reptation theory (33). For obtuse angles, they also perform "hook inversion," a mechanism similar to the one called "jigsaw" (33) or "ratchet" (26,38) in earlier theoretical work. At high fields (14-20 V/cm) (77), another mechanism prevails, dominated by the formation of hernias along the stretched chain as predicted by Deutsch (21,23).

Overall, videomicroscopy provided much more confirmations than contradictions to qualitative theoretical predictions. About the key question of the tube hypothesis, the answer is not clear-cut. Avery general point is that, as expected from entropie considerations, a linear tube seems a rather good description of the chain conformation in low fields (1-3 V/cm, depending on various factors as gel concentration, temperature, and buffer strength), and hernias become increasingly important when the field is increased. These hernias, however, do progress in the gel by a biased reptation mechanism too! So, in our sense, the tube as a representation of topological constraints remains correct at all field strengths (for chains much longer than the pore size). The hypothesis of a linear tube is wrong for high Fields, however, it should be replaced by a model of hernias growing by reptation in a ramified tube. Unfortunately, this is much more difficult to model theoretically!

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